Is "dataset size" and "model size" same thing?
Dataset and model refer to different things. Dataset means part of data available for training (training dataset) or validation (validation dataset). Model is the learning process goal, the state of the computer "brain" after it has been fully educated (or made its learning). Model size refers to size of the container which contains the model. In deep learning it can be measured by width and depth of the network used, I also found a site comparing different models by npy file size, that physically contains the generated model as computer code. In that case model contained a more complex structure which was documented and size in bytes was for comparison purposes.

So in short, it is roughly speaking the size of layers and neurons, if I have to take one of your options. Dataset is a different thing.

More precise explanation about what is model and what is dataset:

https://www.quora.com/What-are-different-models-in-machine-learning